Abstract

The role of the vortical phase in the initial structure of the wave field of a laser beam propagating in the turbulent atmosphere in statistical regularities of beam wandering is studied. It is found that in the near diffraction zone the variances of wandering of the vortical beam and the fundamental Gaussian beam turns out to be identical, if the initial radius of the Gaussian beam is equal to the radius of the ring intensity distribution of the vortical beam. In the far diffraction zone, the vortical beam wanders more slightly than the Gaussian beam with the same effective radius of the initial intensity distribution does. It is also shown that laser beams with the initial ring intensity distribution similar to the intensity distribution of a vortical beam, but not having the vortical phase distribution, are less resistant to the atmospheric turbulence than the vortical beam.

© 2012 Optical Society of America

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References

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  1. H. T. Eyyuboğlu and C. Z. Çil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
    [CrossRef]
  2. V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988).
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
  4. D. C. Cowan and L. C. Andrews, “Effects of atmospheric turbulence on the scintillation and fade probability of flattened Gaussian beams,” Opt. Eng. 47, 026001 (2008).
    [CrossRef]
  5. Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793–3797 (2006).
    [CrossRef]
  6. Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
    [CrossRef]
  7. C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
    [CrossRef]
  8. A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova Science, 2008).
  9. A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
    [CrossRef]
  10. L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
    [CrossRef]
  11. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–229 (2008).
    [CrossRef]
  12. V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
    [CrossRef]
  13. A. I. Kon, V. L. Mironov, and V. V. Nosov, “Dispersion of light beam displacements in the atmosphere with strong intensity fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
    [CrossRef]
  14. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics. Random Fields (Springer, 1987).
  15. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Integrals and Series (Gordon & Breach Science, 1990), Vol. 2.
  16. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).
  17. S. Ramee and R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000).
    [CrossRef]
  18. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086(2003).
    [CrossRef]

2010 (1)

C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
[CrossRef]

2009 (1)

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[CrossRef]

2008 (4)

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–229 (2008).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

H. T. Eyyuboğlu and C. Z. Çil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

D. C. Cowan and L. C. Andrews, “Effects of atmospheric turbulence on the scintillation and fade probability of flattened Gaussian beams,” Opt. Eng. 47, 026001 (2008).
[CrossRef]

2006 (1)

2005 (1)

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

2003 (1)

2000 (1)

1976 (1)

A. I. Kon, V. L. Mironov, and V. V. Nosov, “Dispersion of light beam displacements in the atmosphere with strong intensity fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

1972 (1)

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

Andrews, L. C.

D. C. Cowan and L. C. Andrews, “Effects of atmospheric turbulence on the scintillation and fade probability of flattened Gaussian beams,” Opt. Eng. 47, 026001 (2008).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).

Banakh, V. A.

V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988).

Baykal, Y.

C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793–3797 (2006).
[CrossRef]

Bekshaev, A.

A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova Science, 2008).

Berg, L.

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Integrals and Series (Gordon & Breach Science, 1990), Vol. 2.

Cai, Y.

C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086(2003).
[CrossRef]

Chen, Y.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

Çil, C. Z.

C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
[CrossRef]

H. T. Eyyuboğlu and C. Z. Çil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

Cowan, D. C.

D. C. Cowan and L. C. Andrews, “Effects of atmospheric turbulence on the scintillation and fade probability of flattened Gaussian beams,” Opt. Eng. 47, 026001 (2008).
[CrossRef]

Eyyuboglu, H. T.

C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
[CrossRef]

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

H. T. Eyyuboğlu and C. Z. Çil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793–3797 (2006).
[CrossRef]

Gbur, G.

Klyatskin, V. I.

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Kon, A. I.

A. I. Kon, V. L. Mironov, and V. V. Nosov, “Dispersion of light beam displacements in the atmosphere with strong intensity fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

Korotkova, O.

C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
[CrossRef]

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics. Random Fields (Springer, 1987).

Lin, Q.

Lu, X.

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Integrals and Series (Gordon & Breach Science, 1990), Vol. 2.

Mironov, V. L.

A. I. Kon, V. L. Mironov, and V. V. Nosov, “Dispersion of light beam displacements in the atmosphere with strong intensity fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

Nosov, V. V.

A. I. Kon, V. L. Mironov, and V. V. Nosov, “Dispersion of light beam displacements in the atmosphere with strong intensity fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).

Pokasov, V. V.

V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988).

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Integrals and Series (Gordon & Breach Science, 1990), Vol. 2.

Ramee, S.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics. Random Fields (Springer, 1987).

Simon, R.

Soskin, M.

A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova Science, 2008).

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

Tatarski, V. I.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics. Random Fields (Springer, 1987).

Tyson, R. K.

Vasnetsov, M.

A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova Science, 2008).

Vinotte, A.

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

Wang, L. G.

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[CrossRef]

Zheng, W. W.

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[CrossRef]

Zuev, V. E.

V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988).

Appl. Opt. (1)

Appl. Phys. B (3)

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90, 87–92 (2008).
[CrossRef]

C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0 and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010).
[CrossRef]

H. T. Eyyuboğlu and C. Z. Çil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

J. Opt. A (1)

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

D. C. Cowan and L. C. Andrews, “Effects of atmospheric turbulence on the scintillation and fade probability of flattened Gaussian beams,” Opt. Eng. 47, 026001 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

A. Vinotte and L. Berg, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

Radiophys. Quantum Electron. (2)

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[CrossRef]

A. I. Kon, V. L. Mironov, and V. V. Nosov, “Dispersion of light beam displacements in the atmosphere with strong intensity fluctuations,” Radiophys. Quantum Electron. 19, 722–725 (1976).
[CrossRef]

Other (6)

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics. Random Fields (Springer, 1987).

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Integrals and Series (Gordon & Breach Science, 1990), Vol. 2.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

V. E. Zuev, V. A. Banakh, and V. V. Pokasov, Optics of the Turbulent Atmosphere (Gidrometeoizdat, 1988).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).

A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova Science, 2008).

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Figures (4)

Fig. 1.
Fig. 1.

Normalized rms deviation of the centroid of LG beams as a function of the turbulence parameter D s ( 2 a 0 ) under various conditions of diffraction at the transmitting aperture: l = 0 (curve 1), 4 (2), and 16 (3).

Fig. 2.
Fig. 2.

Square relation σ Cl / σ C 0 of the LG beams with the index l and the Gaussian beam calculated by Eq. (13); the diffraction parameter Ω = 1 ; D s ( 2 a 0 ) = 200 (1), 50 (2), and 0.1 (3).

Fig. 3.
Fig. 3.

Angular dispersion of random displacements of Gaussian (curves 1, 3, and 5, a 0 ( l ) = 0.05 m , l = 0 ) and LG (curves 2, 4, and 6, a 0 = 0.0125 m , l = 16 ) beams as a function of the distance z measured in diffraction lengths z d = k [ a 0 ( l ) ] 2 : C ε 2 = 8.62 · 10 15 m 2 / 3 (curves 1, 2), C ε 2 = 8.62 · 10 17 m 2 / 3 (curves 3 and 4), C ε 2 = 8.62 · 10 19 m 2 / 3 (curves 5 and 6), wavelength λ = 1.065 · 10 6 m .

Fig. 4.
Fig. 4.

Square intensity spectra in the cross section κ y = 0 of the LG (curve 1) and DH (curve 2) beams with l = 16 as functions of the normalized variable κ a 0 x g ( z ) / ( 2 Ω ) . The spectra are normalized to P 0 / 4 π 2 .

Equations (19)

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r C ( z ) = 1 P 0 d 2 r I ( r , z ) · r ,
σ C 2 = 8 π 5 P 0 2 0 z d ξ ( z ξ ) 2 d 2 κ Φ ε ( | κ | , ξ ) κ 2 J ( κ , ξ ) J ( κ , ξ ) ,
J ( κ , z ) = 1 4 π 2 I ( r ; z ) e i κ r d 2 r .
J ( κ ; ξ ) J ( κ ; ξ ) J ( κ ; ξ ) J ( κ ; ξ ) .
σ C 2 = 8 π 5 P 0 2 0 z d ξ ( z ξ ) 2 d 2 κ Φ ε ( | κ | , ξ ) κ 2 J ( κ , ξ ) J ( κ , ξ ) .
u ( r , z ) = 1 a 8 Φ c | l | ! ( r x + i r y a ) | l | exp { r 2 2 a 2 } exp { i r 2 2 F i | l | χ } ,
F ( z ) = z d 2 + z 2 z , a 2 ( z ) = z d 2 + z 2 k z d , χ = arctan ( z / z R ) ,
J ( κ , z ) = J 0 ( κ , z ) exp { π k 2 4 0 z H ( κ z k ( 1 ζ z ) ) d ζ } ,
H ( ρ ) = 2 [ 1 cos κ ρ ] Φ ε ( | κ | , ξ ) d 2 κ .
I 0 ( r , z ) = | u ( r , z ) | 2 = 8 Φ c | l | ! ( ( k z ) Ω g 2 ( z ) ) | l | + 1 r 2 | l | exp { k z Ω g 2 ( z ) r 2 } ,
J 0 ( κ , z ) = 2 Φ c π exp { 1 4 ( g 2 ( z ) Ω ) z k κ 2 } · L | l | 0 ( 1 4 ( g 2 ( z ) Ω ) z k κ 2 ) ,
Φ ε ( | κ | , ξ ) = 0.033 C ε 2 κ 11 / 3 ,
J ( κ , z ) = 2 Φ c π L | l | 0 ( 1 4 a 2 g 2 ( z ) ( z k ) 2 κ 2 ) exp { 1 4 a 2 g 2 ( z ) ( z k ) 2 κ 2 0.142 π · β 0 2 ( z k ) 5 6 κ 5 3 } ,
σ Cl 2 = 0.373 D s ( 2 a ) a 0 2 Ω 11 / 6 0 1 d ξ ( 1 ξ ) 2 0 d κ κ 2 / 3 [ L | l | 0 ( ( ξ 2 + Ω 2 ) 4 Ω κ 2 ) ] 2 × exp { ( ξ 2 + Ω 2 ) 2 Ω κ 2 0.314 D s ( 2 a 0 ) Ω 5 / 6 ξ 8 / 3 κ 5 / 3 } ,
( σ Cl / σ C 0 ) 2 = 1 | l | ! | l | h | l | [ ( 1 h ) 5 / 6 F 1 2 ( 1 12 , 7 12 ; 1 ; h ) ] | h = 0 ,
( σ C l / σ C 0 ) 2 = 2 5 6 2 | l | π · Γ ( 1 6 + 2 | l | ) ( | l | ! ) 2 Γ ( 1 12 ) Γ ( 7 12 ) · F 2 3 ( 5 6 , | l | , | l | ; 5 12 | l | , 11 12 | l | ; 1 ) ,
R ( l ) = | l | · a 0 .
u ( r , 0 ) = 1 a 0 8 Φ c | l | ! ( r a 0 ) | l | exp { r 2 2 a 0 2 } ,
J 0 ( κ , z ) = P 0 2 | l | 1 π 2 | l | ! Γ ( | l | 2 + 1 ) 2 ( Ω a ) 2 ( g 2 ( z ) ) ( | l | 2 + 1 ) × 0 r J 0 ( κ r ) F 1 1 ( | l | 2 + 1 ; 1 ; Ω 2 [ 1 i Ω ] 2 a 2 g 2 ( z ) r 2 ) F 1 1 ( | l | 2 + 1 ; 1 ; Ω 2 [ 1 + i Ω ] 2 a 2 g 2 ( z ) r 2 ) d r .

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